1. Math 478 / 568: Actuarial Modeling
Professor Rick Gorvett
Spring 2015

2. Syllabus
Office Hours: 3-4 pm Tuesdays, 3-4 pm Wednesdays, or by appointment
Textbook: Klugman, Panjer, and Willmot, 4th edition
Exam dates: 3 exams, per syllabus
Grades: Exams, homeworks, project

3. Syllabus (cont.) Graduate Students: Do Math 478, plus an extra project
Project is potentially semester-long
U/G Honors Students: Project alternatives will be handed out ~ half-way through the semester

4. Class Objectives
Understand the mathematical foundations of actuarial modeling
Loss modeling
Model selection and parameter estimation
Credibility theory and simulation
Appreciate this material in a multi-disciplinary context
Learn Exam 4/C material

5. Loss Modeling
How do we represent the potential for financial consequences of events?
Frequency × severity = aggregate loss
Statistical distributions
Frequency – e.g.,
Poisson
Negative binomial
Severity – e.g.,
Lognormal
Exponential
Gamma
Pareto

6. Model Selection and Parameter Estimation
How do we select amongst alternative models and parameters?
How do we use empirical data to determine the characteristics of distributions?
In what sense are some models and parameters “better” or “optimal” in a given situation?

7. Credibility Theory and Simulation
How do we “blend”:
Old and new data?
Group versus individual data?
E.g., { Z•New + (1-Z) •Old }
Simulation
How do we use models to estimate the impact of potential future scenarios?

8. Actuarial Science and Finance
“Coaching is not rocket science.”
- Theresa Grentz, former University of Illinois Women’s Basketball Coach
Are actuarial science and financial mathematics “rocket science”?
Certainly, lots of quantitative Ph.D.s are on Wall Street and doing actuarial-
or finance-related work
But….

9. Actuarial Science and Finance (cont.)
Actuarial science and finance are not rocket science – they’re harder
Rocket science:
Test a theory or design
Learn and re-test until successful
Actuarial science and finance
Things continually change – behaviors, attitudes,….
Can’t hold other variables constant
Limited data with which to test theories

10. Two real-world examples
Motivation
Two real-world examples

11. Space Shuttle Challenger Explosion
Example # 1
Space Shuttle Challenger Explosion

12. Facts Leading Up to Launch…
23 successful launches prior to January 28, 1986
Previous launches at temperatures from 53°F to 81°F
Challenger launch on morning of 1/28/86 was at 31°F – far below previous launches

13. Launch / O-Ring Information
Launch vehicle configuration:
Orbiter
External fuel tank
Two solid rocket boosters, manufactured by Morton Thiokol (MT)
Sections sealed with O-rings, whose performance is sensitive to temperature
But: MT’s recommendation stated that “Temperature data (are) not conclusive on predicting primary O-ring blowby.”

14. The Result Vehicle exploded 73 seconds after launch
Cause (per Rogers Commission): gas leak in SRB, caused by failure or degredation of O-ring, led to weakening or penetration of external fuel tank
Rogers Commission conclusion: “A careful analysis of the flight history of O-ring performance would have revealed the correlation of O-ring damage in low temperature.”

15. Statistical Analysis
How predictable was it?
Data:

16. Statistical Analysis (cont.)
Or:
Charts from “Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of Failure,” by Dalal, et al, Journal of the American Statistical Association, December 1989

17. Taco Bell and The Mir Space Station
Example # 2
Taco Bell and The Mir Space Station

18. Taco Bell and Mir Space Station Mir Taco Bell In orbit for 15 years
Expected to crash back to earth on March 24, 2001, in the Pacific Ocean
Size of projected debris field: 200 km × 6,000 km
Taco Bell
Offered a free Crunchy Beef Taco to every U.S. resident if the core of Mir hit a 144 square-meter target 15 km off Australian coast

20. Taco Bell and Mir (cont.)
Suppose you are an actuary, working for an insurance firm
Your firm has been approached by Taco Bell to insure against the potential financial loss associated with their possible Mir-related payout
What’s a reasonable price for such coverage?

21. Taco Bell and Mir (cont.)
Aggregate loss = frequency times severity
What is the probability of Mir hitting the target?
What will it cost Taco Bell if it does?

22. Taco Bell and Mir (cont.)
Potential cost:
Population of United States?
Cost of a Crunchy Beef Taco?
Potential cost?
Probability of a hit?
Indicated premium?

23. Taco Bell and Mir (cont.)
Issues:
Uniform distribution across debris field?
How many will cash in?
What about expenses / fixed costs?

24. Next Time Random variables
Conditional probabilities and Bayes Theorem.
Survival functions.
Hazard rates.